Fig. 3: Parallel implementation of a universal set of logical gates by code concatenation.

A logical qubit is encoded in the [[7, 1, 3]] Steane code concatenated with the [[15, 1, 3]] QRM code forming a 7 × 15 atom array (dark blue circles) held by an array of static tweezers generated with a spatial light modulator (SLM, red circles), where each row is a 15-qubit code and a physical “qubit” for the 7-qubit code. To apply CNOT gates transversally between two rows, say rows 1 and 6, one can use an array of mobile tweezers generated with AOD to shift one row of atoms to the neighbor sites of the other row (green arrow). One then turns off the tweezers, applies a row of Hadamard gates on the target line, turns on a pulse to apply CZ gates on the atom pairs closer than the Rydberg blockade radius simultaneously, and applies Hadamard gates on the target line again. To apply physical CNOT gates simultaneously between two columns, say columns 5 and 9, one can move one column of atoms to the neighbor of the other (purple arrow). The other operations are similar.